# Systems the most widely used control algorithms

Systems and Controls 2: Quanser Helicopter Lab
Andreas-Marios Giompazolias-Schoinas,Student ID:ag16408,Candidate Number:40189
Ioannis Girousis,Student ID:ig16494,Candidate Number:37998
January 31, 2018
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1 Introduction to PID controls
Proportional Integral Derivative (PID) controller is the most widely used control loop algorithm in industry, as
more than 95 % of all the regulatory controller utilize PID feedback system. Based on the heuristic approach
feedback controllers allow for the response to be sufficiently close to the desired value and are therefore ideal for
most practical control problems.
A control feedback system was first established in the industry by James Watt’s “conical pendulum” governor.
This control system was operating with only a proportional gain (Kp) resulting in wider oscillations of the Process
Variable for bigger values of the Kp or higher load.
The history of PID controllers in the form that we know them today started on 1868 when James Clerk Maxwell
explored the mathematical basis of controlling systems stability in its’ famous paper “On Governors”. In the same
year, a pendulum-and-hydro-stat control was introduced in the newly invented Whitehead torpedo being one of the
first PID control systems to be used in the industry. The last step of defining the PID controllers came from Nicolas
Minorsky, who presented a formal control law using theoretical analysis. Since then PID controllers are spreading
in the industry becoming until today the most used control loop in the industry.
Figure 1: Step function
A step function as seen above, can be used to measure control system performance requirements representing
the set point. Then by measuring the process variable response over time and using waveform characteristics
(represented in figure 1 above) such as, rise time, overshoot and steady-state error the response is evaluated.
Feedback control allows the correction of errors in a system based on the desired and actual output. One of
the most widely used control algorithms is the Proportional Integral Derivative controller, thanks to its effective
performance in a variety of operating conditions and its simplicity. A PID consists of 3 coefficients the values
of which can be varied in order to achieve the optimal response. The PID controller can be represented by the
following equation where, u(t) is the control signal and e(?) the error, which is the difference between the desired
result and the actual response of the system.
u(t) = KP(t) + KI
Z 0
t
e(?)d? + KD
de
dt
(1)
The control signal is comprised of 3 terms, proportional, integral and derivative, with control parameters kp, ki
and kd respectively. The above equation can also be expressed in the Laplace domain, where the roles of each term
is better represented:
GR(s) =
U(s)
E(s)
= KP(1 +
1
TIs
+
TDs
1 +
TD
N
× s
) (2)
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In its Laplace form it easier to manipulate and and relate to practical results, especially in higher order problems.
Figure 2: PID controller block diagram
For the proportional component, the error magnitude is multiplied by the proportional gain Kp resulting in the
proportional response. Therefore the greater the percentage difference of the output signal from the desired output
is, the greater the proportional response will be.
Integral:
KP × e(t) (3)
The integral component sums the error term over time, multiplying it by the integral coefficient Ki and then
integrating.Even a minor deviation from the desired value will result in a large integral component over time. The
steady-state errors, the final difference between the process variable and set point, will be driven to zero through
the integral component since the integral response will rise until the error is zero.
Proportional:
KI ×
Z t
tD
e(t) (4)
The derivative response is proportional to the rate of change of the process variable. Hence the control system
is able to minimize the effect of a disturbance and the time it takes to correct one, through the use of a projection of
the process variable value. This allows higher Integral and Proportional coefficients to be used without the system
becoming unstable.However since any changes in the error term affect it, the component is highly sensitive to noise
in the sensor feedback and can easily render the system unstable if it is set too large.
Derivative:
KD ×
d
dt
e(t) (5)
2 Block Diagram
The effects of a corresponding increase at each individual component of the PID are represented below in Table 1.
The term rise time refers to the time taken for the helicopter to rise from 10% to 90% of its steady value. Overshoot
is when the helicopter goes past the required elevation, desired response signal. Settling time is the time period
for the helicopter to remain within a certain margin from the required elevation with no further deviations. Steady
State Error (SSE) is the difference between the desired result and the actual response of the system, u(t) ? e(t). The
term stability refers to the oscillatory behavior of the system.
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Table 1: Increasing the independent variable values of a PID control
Parameter Rise time Overshoot Settling Time Steady State Error Stability
KP Decrease Increase Slight Change Decrease Degrade
KD Slight Change Decrease Decrease No effect in theory Improve if KD is small
3 Qunaser Helicopter PID control practical results
In this part of the laboratory, the values of KP, KD, KI where tested individually, by adjusting their values and
observing the visual result on the signal. The overall results, satisfy our theoretical background.
Figure 3: Test values for KP, a:0.3, b:0.5, c:0.7, d:0.9
When increasing the value of KP we can see that the overall Rise time decreases, the Overshoot increases and
there is no particular change Settling time. However, having a high KP results to instabilities, as can be seen from
Figure 3: d (KP=KPi + 0.4). Furthermore, the benefits than can be obtained by rising KP can be achieved by
to a higher value too. Therefore, increasing Kp does not ensures us with a gradual and smooth result
in steady state. As a result, the final value is decreased compared to the initial one, Figure 3: b (KPi = 0.5), to
KP f = 0.27, to avoid unwanted oscillations.
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Figure 4: Test values for KI
, I:0.03, II:0.08, III:0.18, IV:0.28
Adjusting KI variable to a higher value, as can be seen from the above Figure, the Rise time and the Steady
State Error decreases, while the Overshoot and the Settling time increase. Moreover, the change of the value KI
, as
it is demonstrated in Figure 4, doesn’t affect the stability greatly, like KP. Consequently, the value of KI has been
raised to KI = 0.23 from an initial value of KIi = 0.08
Figure 5: Test values for KD, A:0.20, B:0.34, C:0.40, D:0.48
Altering the value of KD in this the overshoot and the Settling Time. Raising KD, gives smaller overshoot and
shorter Settling Time with no visual effect on Rise Time or Steady State Error. As a result, the final value of KD is
0.40, +0.06 from the initial value.
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4 Discussion
In order to achieve the desired response some tuning had to be undertaken, adjusting the proportional, differential
and integral gains. For that purpose a trial and error method was followed. A 0.2 , 0.07 and 0.01, step was used
for each of KP,KD and KI respectively, hence finding the optimum value for each component and balance between
them.The progress can be seen in Figures 3 to 6, as each component is adjusted the rise time, overshoot and settling
time are stabilized and the signal deviation from the ideal response is minimized. As seen in Figure 6 the final
values were +0.15 for KI
, ?0.23 for KP and +0.06 for KD.
Figure 6 Final Results
: KI + 0.15, KP ? 0.23, KD + 0.06
5 References
References
1 DesboroughHoneywell.PIDControl.”Caltech.edu, 2000, pp.301–322.,
www.cds.caltech.edu/ murray/books/AM08/pd f /am06 ? pid16S ep06.pd f.
2 PID f orDummies.”PID f orDummies?ControlS olutions,www.csimn.com/CS Ipages/PID f orDummies.html.
3 “PID Theory Explained.” PID Theory Explained – National Instruments, www.ni.com/white-paper/3782/en/.
4 Smuts, Jacques. “PID Controllers Explained.” Blog.opticontrols, 7 Mar. 2011,
blog.opticontrols.com/archives/344.
5 Smuts, Jacques F. Process Control for Practitioners: How to Tune PID Controllers and Optimize Control Loops.
OptiControls, 2011.
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